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Chapter 10: Problem 28

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=10^{x}, a=2$$

Short Answer

Expert verified
Answer: The first four nonzero terms of the Taylor series are \( T_4(x)= 100 + (100\ln(10))(x-2) +\frac{100(\ln(10))^2}{2!}(x-2)^2 +\frac{100(\ln(10))^3}{3!}(x-2)^3 \) and its summation notation representation is \( T_4(x)=\sum_{n=0}^3\frac{100(\ln(10))^n}{n!}(x-2)^n \).

Step by step solution

01

Find the first four derivatives of the function

To find the first four derivatives of the function \(f(x)=10^x\), we will rewrite \(10^x\) as \(e^{x\ln(10)}\), and then find the derivatives. 1. First derivative: \(f'(x)=e^{x\ln(10)}\ln(10)\) 2. Second derivative: \(f''(x)=e^{x\ln(10)}(\ln(10))^2\) 3. Third derivative: \(f'''(x)=e^{x\ln(10)}(\ln(10))^3\) 4. Fourth derivative: \(f^{(4)}(x)=e^{x\ln(10)}(\ln(10))^4\)
02

Evaluate the derivatives at a = 2

Now we need to evaluate the four derivatives at \(a=2\): 1. \(f(2)=10^2=100\) 2. \(f'(2)=e^{2\ln(10)}\ln(10)=100\ln(10)\) 3. \(f''(2)=e^{2\ln(10)}(\ln(10))^2=100(\ln(10))^2\) 4. \(f'''(2)=e^{2\ln(10)}(\ln(10))^3=100(\ln(10))^3\)
03

Write the first four non-zero terms of the Taylor series

Now, we can substitute the values of the derivatives evaluated at \(a=2\) into the Taylor series formula: $$T_4(x) = 100 + (100\ln(10))(x-2) +\frac{100(\ln(10))^2}{2!}(x-2)^2 +\frac{100(\ln(10))^3}{3!}(x-2)^3$$
04

Write the power series in summation notation

We can now write the obtained power series in summation notation: $$T_4(x)=\sum_{n=0}^3\frac{100(\ln(10))^n}{n!}(x-2)^n$$ Now we have found the first four nonzero terms of the Taylor series for the given function centered at \(a=2\) and written the power series using summation notation.

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Most popular questions from this chapter

Given the power series $$\frac{1}{\sqrt{1-x^{2}}}=1+\frac{1}{2} x^{2}+\frac{1 \cdot 3}{2 \cdot 4} x^{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{6}+\cdots$$ for \(-1< x <1,\) find the power series for \(f(x)=\sin ^{-1} x\) centered at \(0 .\)

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k}$$

Evaluating an infinite series Let \(f(x)=\left(e^{x}-1\right) / x,\) for \(x \neq 0\) and \(f(0)=1 .\) Use the Taylor series for \(f\) about 0 and evaluate \(f(1)\) to find the value of \(\sum_{k=0}^{\infty} \frac{1}{(k+1) !}\)

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$$

Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is $$J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k}.$$ a. Write out the first four terms of \(J_{0}\) b. Find the radius and interval of convergence of the power series for \(J_{0}\) c. Differentiate \(J_{0}\) twice and show (by keeping terms through \(x^{6}\) ) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\)
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